# Computational methods for the discrete downward continuation of the earth gravity and effects of lateral topographical mass density variation on gravity and the geoid

## Abstract

In this thesis, computational methods for the discrete Poisson downward continuation of the Earth gravity are studied. In addition, the effect of the lateral topographical mass density variation on gravity and the geoid is systematically investigated.
A solution in the spherical harmonic form for the Poisson integral equation is derived. It is pointed out that the solution of the discrete inverse Poisson problem exists, but may not be unique and stable. For a small input error, a larger error is introduced. It is in this sense that the inverse Poisson problem is said to be an ill-posed problem.
It is found that the modified spheroidal Poisson kernel reduces the ‘real’ far-zone contribution with respect to using the spheroidal Poisson kernel significantly, but it cannot perform better than the standard Poisson kernel in reducing it. A fast algorithm is developed for the evaluation of the far-zone contribution.
Heiskanen and Moritz’s (1967) radius condition gives a critical radius of the near-zone cap that is too small for the determination of the em-geoid, while Martinee’s (1996) condition gives an unnecessarily large radius. It is proposed that the critical near-zone radius be determine as a function of the accuracy of the global geopotential model from which the far-zone contribution is evaluated.
The combined iterative method is proposed to speed up the convergence of the solution of the discrete inverse Poisson problem. The truncated singular value decomposition method is introduced to solve the discrete Poisson integral equation that may be ill-conditioned for a small discrete step.
The three discrete models for the Poisson integral, namely the point-point, point-mean and mean-mean models are assessed against synthetic data. It is shown that the mean-mean model can produce a sufficiently accurate solution when the so called ‘averaging error’ is properly corrected for.
A block-wise technique is developed to solve the discrete Poisson integral equation efficiently with a compressed storage technique for the coefficient matrix. The package DOWN97 is developed to evaluate the discrete Poisson downward continuation.
The analytical downward continuation is compared to the Poisson one by using synthetic data and Helmert gravity anomalies. It is shown that the analytical downward continuation agrees with the discrete Poisson’s within 10% of the total downward continuation effect. A fast algorithm is developed to evaluate the analytical downward continuation with its implementation in software.
A ultra-high degree synthetic global geopotential model conforming to the Kaula-type degree variance models is too smooth to simulate the Helmert gravity anomalies, especially in mountainous regions.
An approach for evaluating the effect of the lateral topographical mass density variation on gravity and the geoid is proposed. Formulae for evaluating the standard deviation of the primary indirect topographical mass density effect (PIDE), the direct topographical mass density effect (DDE) and the secondary topographical mass density effect (SIDE) are derived. The effect of the lateral topographical mass density variation on the geoid height ranges from -7.0cm to 2.8 cm by using a 30” X 30” DTM (Digital Topographic Model). The software for evaluating the PIDE, DDE and SIDE is described.